$f(x, y) = (5 + x, y\ln(x + y))$ What is the curl of $f$ at $(4, 8)$ ?
The formula for curl in two dimensions is $\text{curl}(f) = \dfrac{\partial Q}{\partial x} - \dfrac{\partial P}{\partial y}$, where $P$ is the $x$ -component of $f$ and $Q$ is the $y$ -component. Let's differentiate! $\begin{aligned} \dfrac{\partial Q}{\partial x} &= \dfrac{\partial}{\partial x} \left[ y \ln(x + y) \right] \\ \\ &= \dfrac{y}{x + y} \\ \\ \dfrac{\partial P}{\partial y} &= \dfrac{\partial}{\partial y} \left[ 5 + x \right] \\ \\ &= 0 \end{aligned}$ Therefore: $\text{curl}(f) = \dfrac{y}{x + y}$ The curl of $f$ at $(4, 8)$ is $\dfrac{2}{3}$.